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Huzefa Kapadia, an 800-scoring math specialist, breaks SAT Math into fast, high-leverage strategies and clear walkthroughs of the trickier question types.
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Math · Checkpoint 17 of 30 · Advanced Math · ~15 min
Quadratics & polynomials
Factoring, roots, the vertex, and reading parabolas.
The 30-second version
a quadratic has three costumes, and each reveals one secret instantly. Standard y = ax² + bx + c shows the y-intercept (c). Factored y = a(x − p)(x − q) shows the roots (p and q). Vertex y = a(x − h)² + k shows the max/min point (h, k). Half of SAT quadratic questions are just asking you to change costumes or read the right secret.
Learn it
Factoring, fast
x² + 7x + 12: two numbers that multiply to 12 and add to 7 → 3 and 4 → (x + 3)(x + 4). Set each factor to zero for roots: x = −3, −4. Know the two special patterns on sight: difference of squares a² − b² = (a + b)(a − b), and perfect squares a² ± 2ab + b² = (a ± b)².
The vertex
The parabola's turning point. From standard form, its x-coordinate is x = −b/2a; plug back in for y. "What is the maximum height of the projectile?" = "find the vertex's y." "After how many seconds?" = "find the vertex's x." If the equation is already in vertex form, the answer is sitting there: y = −2(x − 3)² + 50 peaks at (3, 50).
How many solutions? The discriminant
For ax² + bx + c = 0, compute b² − 4ac: positive → two real solutions, zero → exactly one (the parabola kisses the x-axis), negative → none. "For what k does the equation have exactly one solution?" → set b² − 4ac = 0 and solve for k.
Polynomials beyond quadratics
Two facts cover nearly everything: (x − a) is a factor exactly when plugging in a gives zero, and a graph's x-intercepts are the polynomial's real roots (touching-without-crossing means a repeated root).
Common traps
- Sign flips in forms: (x − 3) means root at +3; vertex form (x − h) means h is +. The minus sign is part of the template.
- Solving x² = 25 and forgetting −5. Square roots in equations come in pairs.
- Answering a root when the question wants the sum or product of roots. (Sum = −b/a, product = c/a — worth memorizing.)
Try it
The height of a ball is h(t) = −(t − 4)² + 36. What is the ball's maximum height?
Vertex form: vertex at (4, 36), opening downward (negative out front), so the maximum height is 36 — reached at t = 4 seconds. A is the time of the peak, the classic wrong answer for reading the wrong coordinate.
Free resources for this checkpoint
LearnSATMath (opens in a new tab)
A math-only channel with the tagline 'only the essentials' — concise lessons and question walkthroughs focused specifically on SAT Math skills.
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Practice next
Question Bank filter: Math → Advanced Math → Nonlinear equations and Nonlinear functions. Quadratics are also where Desmos starts paying off hugely — graph one and tap the vertex.
Practice Equivalent expressionsOriginal Digital SAT-style questions on what this checkpoint covers.